434 
MR. 0. HEAVISIDE ON THE FORCES, STRESSES, AND 
-6 = i (IA 3 — P 32 ) + J (P 31 — P 13 ) + ^ (P 12 — ^*21 )• • • 
(37) 
Here 26 is the torque per unit volume arising from the stress P. 
The translational force, F, per unit volume is (by inspection of a unit cube) 
F — + V 2 P 2 4- V 3 P 3 .(38) 
= i div dj + j div CD + k div Q, 3 ;.(39) 
or, in terms of the self-conjugate stress and the torque, 
F = (i div 0 o i + j div 0 o j + k div 0 o k) — curl €,.(40) 
where — curl € is the translational force due to the rotational stress alone, as in Sir 
W. Thomson’s latest theory of the mechanics of an “ ether,”* 
Next, let N be the unit normal drawn outward from any closed surface. Then 
2p n = 2f, .• . . . . (41) 
where the left summation extends over the surface and the right summation through¬ 
out the enclosed region. For 
P N = NfP, + N 2 P 2 + N,P S 
= i.NCfi + j.Ndo + k.NQ,;.(42) 
so the well-known theorem of divergence gives immediately, by (39), 
2P N = 2 (i div Q, 1 -f j div CD + k div Q 3 ) = 2 F.(43) 
Next, as regards the equivalence of rotational effect of the surface-stress to that of 
the internal forces and torques. Let r be the vector distance from any fixed origin. 
Then VrF is the vector moment of a force, F, at the end of the arm r. Another 
(not so immediate) application of the divergence theorem gives 
2VrP N = 2 VrF + 226, .(44) 
Thus, any distribution of stress, whether rotational or irrotational, may be regarded as 
in equilibrium. Given any stress in a body, terminating at its boundary, the body 
will be in equilibrium both as regards translation and rotation. Of course, the 
boundary discontinuity in the stress has to be reckoned as the equivalent of internal 
divergence in the appropriate manner. Or, more simply, let the stress fall off 
continuously from the finite internal stress to zero through a thin surface-layer. We 
* ‘ Mathematical and Physical Papers,’ vol. 3, Art. 99, p. 436. 
